Galileo, according to
his perhaps-unreliable biographer Viviani,
timed the period of the chandeliers in Pisa Cathedral,
and dropped weights from the Leaning Tower.

2013 April 29:

The principle of equivalence -- in its "weak" version -- was announced by Galileo in 1638:
heavy stones and light fall with the same acceleration, as would all bodies in the
absence of air resistance. The principle could be tested by dropping heavy objects,
by rolling balls down inclined planes, by observing the regular swing of
pendula ...

Chandelier in Pisa Cathedral.
Photo by Katarina Jankovic.

This was a cutting-edge research-area throughout
the Seventeenth Century, as gravitational theory developed into its Cartesian
and then its Newtonian forms.
Newton himself conducted an ingenious
experiment in the Galilean tradition:[Principia III, vi, 6]:

"It has been, now of a long time, observed by others, that all sorts of
heavy bodies (allowance being made for the inequality of retardation which
they suffer from a small power of resistance in the air) descend to the
earth from equal heights in equal times; and that equality of times we
may distinguish to a great accuracy, by the help of pendulums.

"I tried the
thing in gold, silver, lead, glass, sand, common salt, wood, water, and wheat.
I provided two wooden boxes, round and equal : I filled the one with wood,
and suspended an equal weight of gold (as exactly as I could) in the centre
of oscillation of the other. The boxes hanging by equal threads of 11 feet
made a couple of pendulums perfectly equal in weight and figure, and
equally receiving the resistance of the air. And, placing the one by the
other, I observed them to play together forward and backward, for a long
time, with equal vibrations. And therefore the quantity of matter in the
gold (by Cor. 1 and 0, Prop. XXIV, Book II) was to the quantity of
matter in the wood as the action of the motive force upon all
the gold to the action of the same upon all the wood ; that is, as the weight
of the one to the weight of the other : and the like happened in the other
bodies. By these experiments, in bodies of the same weight, I could
manifestly have discovered a difference of matter less than the thousandth part
of the whole, had any such been."

From this experiment Newton derived two much stronger statements of the
equivalence principle than Galileo's:

"Corollary 1. Hence the weights of bodies do not depend upon their forms
and textures ; for if the weights could be altered with the forms, they
would be greater or less, according to the variety of forms, in equal matter ;
altogether against experience.

"Corollary 2. Universally, all bodies about the earth gravitate towards the
earth ; and the weights of all, at equal distances from the earth's centre,
are as the quantities of matter which they severally contain. This is the
quality of all bodies within the reach of our experiments ; and therefore
(by Rule III) to be affirmed of all bodies whatsoever. If the æther, or any
other body, were either altogether void of gravity, or were to gravitate less
in proportion to its quantity of matter, then, because (according to
Aristotle, Des Cartes, and others) there is no difference betwixt that and other
bodies but in mere form of matter, by a successive change from form to
form, it might be changed at last into a body of the same condition with
those which gravitate most in proportion to their quantity of matter ; and,
on the other hand, the heaviest bodies, acquiring the first form of that
body, might by degrees quite lose their gravity. And therefore the weights
would depend upon the forms of bodies, and with those forms might be
changed : contrary to what was proved in the preceding Corollary."

I had expected to find without difficulty
a short video showing that simple pendula with different
masses have the same period. Amazingly, very few
(excluding simulations) seem to be available;
most of those I found, like this one, are obviously student lab-reports.
[1 min]

Newton's version of the pendulum experiment typifies how scientific
replication usually works in the context of an ongoing research-programme.
One reads an account of some fairly recent experiment
(in Newton's day meant "recent" meant less than a century old; today it
might mean weeks) but does not attempt to exactly copy it
as given in the literature; rather, one designs a related experiment
testing the predictions of the first experiment by a slightly different method.
Newton did not travel to Pisa, or even to a domed building in
England, and measure the periods of chandeliers; he constructed
small pendula of a standard length but bobs of various
weights and compositions.
Had this experiment yielded a wide variety of different periods,
the validity of Galileo's observations would have been called into
question, and a visit to a cathedral might have been warranted (to investigate
the possibility that only very long pendula have a mass-independent
period). Instead, of course, Newton's results agreed with what
Galileo might have predicted, and also added some additional
information.

Although the intention here was clearly not to conduct an
historical investigation, one might nevertheless describe such
replications as cenochronic: one is checking, and
also extending, the work of a contemporary researcher.

In modern retellings, the equivalence principle of Galileo
and Newton is usually given as:

Weak Form: weight is proportional to quantity of matter;
the "mass" that appears in the law of gravity is the same
as the "mass" in the law of inertia. If mass is the ratio of
force to acceleration and weight (i.e. gravitational force)
is the product of mass and gravitational field, then:

Stronger Form: the gravitational field
is the acceleration due to gravity.
Thus the gravitational response of an object
does not depend on its mass or any other physical properties
(Newton's "forms and textures"), but only on its location in the
field (and perhaps its velocity).

The gravitational field
is acceleration; note the
copula. In all of the 1700s -- in all of the
1800s -- noöne seems to have thought to
reverse the sentence and say that acceleration is
a gravitational field. Noöne, that is, except
one mathematician of medium repute, far better known as an author of
children's novels ... not so much of this
novel, however. Sylvie and Bruno takes place in at least three
different universes, parallel yet intersecting in non-Euclidean
fashion; the part which concerns us here plays out
in ours, among the familiar rituals of a Victorian country-house.

Here is Lewis Carroll, trying
his best to write like Austen or Trollope as he documents the
bumpy course of true love among the gentry ... Naturally,
there are some distractions besides the usual ones:

" 'How convenient it would be,' Lady Muriel
laughingly remarked, à propos of my having
insisted on saving her the trouble of carrying a
cup of tea across the room to the Earl, 'if
cups of tea had no weight at all ! Then perhaps
ladies would sometimes be permitted to
carry them for short distances !'

" 'One can easily imagine a situation,' said
Arthur, 'where things would necessarily have
no weight, relatively to each other, though each
would have its usual weight, looked at by
itself.'

" 'Some desperate paradox !' said the Earl.
'Tell us how it could be. We shall never
guess it.'

" 'Well, suppose this house, just as it is,
placed a few billion miles above a planet, and
with nothing else near enough to disturb it : of
course it falls to the planet ?'

"The Earl nodded. 'Of course -- though it
might take some centuries to do it.'

" 'And is five-o'clock-tea to be going on all
the while ? ' said Lady Muriel.

" 'That, and other things,' said Arthur. 'The
inhabitants would live their lives, grow up and
die, and still the house would be falling, falling,
falling ! But now as to the relative weight of
things. Nothing can be heavy, you know,
except by trying to fall, and being prevented
from doing so. You all grant that ? '

"We all granted that.

" 'Well, now, if I take this book, and hold it
out at arm's length, of course I feel its weight.
It is trying to fall, and I prevent it. And, if I
let go, it falls to the floor. But, if we were all
falling together, it couldn't be trying to fall any
quicker, you know : for, if I let go, what more
could it do than fall ? And, as my hand would
be falling too -- at the same rate -- it would
never leave it, for that would be to get ahead of
it in the race. And it could never overtake the
falling floor ! '

" 'I see it clearly,' said Lady Muriel. 'But
it makes one dizzy to think of such things !
How can you make us do it ?'

" 'There is a more curious idea yet,' I ventured
to say. 'Suppose a cord fastened to the
house, from below, and pulled down by some
one on the planet. Then of course the house
goes faster than its natural rate of falling : but
the furniture -- with our noble selves -- would
go on falling at their old pace, and would
therefore be left behind.'

" 'Practically, we should rise to the ceiling,'
said the Earl. 'The inevitable result of which
would be concussion of brain.'

" 'To avoid that,' said Arthur, 'let us have
the furniture fixed to the floor, and ourselves
tied down to the furniture. Then the five-o'clock-tea
could go on in peace.'

" 'With one little drawback!' Lady Muriel
gaily interrupted. 'We should take the cups
down with us : but what about the tea?'

" 'I had forgotten the tea,' Arthur confessed.
'That, no doubt, would rise to the ceiling-
unless you chose to drink it on the way !'

" 'Which, I think, is quite nonsense enough for
one while !' said the Earl. 'What news does this
gentleman bring us from the great world of London ?'

"... [T]he conversation ... now took a more
conventional tone ..."

A more conventional tone indeed. Carroll/Dodgson
in these paragraphs had done better and deeper physics
than Ernst Mach and his entire over-rated
school, but noöne in the scientific world
seems to have paid the least attention!
(It's hard to blame them too much, though -- reading
Sylvie and Bruno can be an ordeal:
" 'It are gone !' Bruno solemnly replied ... 'Oo couldn't touch it, oo know.
If oo walked at it, oo'd go right froo!'")

"THE EQUALITY OF INERTIAL AND GRAVITATIONAL
MASS AS AN ARGUMENT FOR THE
GENERAL POSTULATE OF RELATIVITY

"We imagine a large portion of empty space, so far
removed from stars and other appreciable
masses, that we have before us approximately
the conditions required by the fundamental law of Galilei.
It is then possible to choose a Galileian reference-body for
this part of space (world), relative to which points at
rest remain at rest and points in motion continue
permanently in uniform rectilinear motion. As reference-body
let us imagine a spacious chest resembling a room
with an observer inside who is equipped with apparatus.
Gravitation naturally does not exist for this observer.
He must fasten himself with strings to the floor,
otherwise the slightest impact against the floor will
cause him to rise slowly towards the ceiling of the
room.

"To the middle of the lid of the chest is fixed externally
a hook with rope attached, and now a 'being' (what
kind of a being is immaterial to us) begins pulling at
this with a constant force. The chest together with the
observer then begin to move 'upwards' with a
uniformly accelerated motion. In course of time their
velocity will reach unheard-of values provided that
we are viewing all this from another reference-body
which is not being pulled with a rope.

"But how does the man in the chest regard the process ?
The acceleration of the chest will be transmitted to him
by the reaction of the floor of the chest. He must
therefore take up this pressure by means of his legs if
he does not wish to be laid out full length on the floor.
He is then standing in the chest in exactly the same way
as anyone stands in a room of a house on our earth.
If he release a body which he previously had in his
hand, the acceleration of the chest will no longer be
transmitted to this body, and for this reason the body
will approach the floor of the chest with an accelerated
relative motion. The observer will further convince
himself that the acceleration of the body towards the floor
of the chest is always of the same magnitude, whatever
kind of body he may happen to use for the experiment.

"Relying on his knowledge of the gravitational field
(as it was discussed in the preceding section), the man
in the chest will thus come to the conclusion that he
and the chest are in a gravitational field which is constant
with regard to time. Of course he will be puzzled for
a moment as to why the chest does not fall, in this
gravitational field. Just then, however, he discovers
the hook in the middle of the lid of the chest and the
rope which is attached to it, and he consequently comes
to the conclusion that the chest is suspended at rest in
the gravitational field.

"Ought we to smile at the man and say that he errs
in his conclusion ? I do not believe we ought to if we
wish to remain consistent ; we must rather admit that
his mode of grasping the situation violates neither reason
nor known mechanical laws. Even though it is being
accelerated with respect to the 'Galileian space'
first considered, we can nevertheless regard the chest
as being at rest. We have thus good grounds for
extending the principle of relativity to include bodies
of reference which are accelerated with respect to each
other, and as a result we have gained a powerful argument
for a generalised postulate of relativity ...

" ... Suppose that the man in the chest fixes a rope to the
inner side of the lid, and that he attaches a body to the
free end of the rope. The result of this will be to stretch
the rope so that it will hang 'vertically' downwards.
If we ask for an opinion of the cause of tension in the
rope, the man in the chest will say : 'The suspended
body experiences a downward force in the gravitational
field, and this is neutralised by the tension of the rope ;
what determines the magnitude of the tension of the
rope is the gravitational mass of the suspended body.'

"On the other hand, an observer who is poised freely in
space will interpret the condition of things thus : 'The
rope must perforce take part in the accelerated motion
of the chest, and it transmits this motion to the body
attached to it. The tension of the rope is just large
enough to effect the acceleration of the body. That
which determines the magnitude of the tension of the
rope is the inertial mass of the body.' Guided by
this example, we see that our extension of the principle
of relativity implies the necessity of the law of the
equality of inertial and gravitational mass. Thus we
have obtained a physical interpretation of this law."

Einstein in his Berlin years [Scientific Monthly 10, 418 (1920)]

Obviously both Einstein and Dodgson were conducting
thought experiments; nonetheless, it would be correct
to say that they were (mentally) re-creating the work
of Galileo and of Newton, and doing so at a very high level.
It should be emphasised that the equivalence principle, in
its Newtonian form, was no longer a research topic in the
1880s or the 1900s; it was standard, textbook material.
Although there were numerous real-world experiments
descended from Galileo's and conducted by researchers,
their purpose was not to shed light upon the nature of
mass or gravity, but to use the established results to
(for example) "weigh the Earth", add more digits to
the constant G, or locate buried geological
structures by their gravitational signatures. In such
experiments and surveys, the equivalence principle was
taken for granted, as something long known. In a sense,
then, these could be described as "anachronic" replications.

In devising the thought experiments which led to general relativity, by contrast,
Dodgson and Einstein asked difficult questions about the relationship
between acceleration, mass, and gravity -- the same questions
that Newton had asked, and that Galileo had asked to the extent
that his vocabulary permitted. For Dodgson and Einstein,
Newton and Galileo were not departed ancestors (to be venerated
or forgotten) but contemporaries across time. By (mentally) repeating
the classic falling-body experiments, a new insight was obtained,
one already partly latent in the older work: acceleration and gravity
cannot be distinguished
at all. This is a cenochronic insight.

Readers of this blog-post may also wish to read
"
Einstein's Pathway to the Equivalence Principle"
by Galina Weinstein.
Next week we will see how cenochronic re-creations of
Galileo's experiment in the laboratory have become an
ongoing part of modern gravitational research. But for now ---

"'How perfectly isochronous!' the Professor exclaimed with enthusiasm. He had his watch in his hand, and was
carefully counting Bruno's oscillations. 'He measures time quite as accurately as a pendulum!'

" 'Yet even pendulums,' the good-natured young soldier observed, as he carefully released his hand from Bruno's grasp,
'are not a joy for ever! Come, that's enough for one bout, little man!'"